A semipositive holomorphic line bundle, in the smooth Hermitian sense, is one admitting a smooth Hermitian
metric with nonnegative Chern curvature. The
curvature is a real 
-form. Smooth semipositivity implies nefness, while nefness
allows approximate lower curvature bounds with metrics depending on the approximation
parameter. A singular semipositive metric instead has plurisubharmonic local weights
and may have unbounded singularities.
The existence of one such metric is stronger input than numerical nefness alone.
